The amount of mass in each volume should be the same if they have the same size.

And you are still unjustifiably assuming a uniform distribution.

But like any good scientist you must account for the possibility that your assumption is false and consider if any evidence supports the alternative.

The 2nd law of thermodynamics says entropy increases with time so if the heat death of the universe is maximum entropy (e.g uniform distribution of matter through the volume of the universe) then at minimum entropy (e.g Big Bang) matter needs not be uniformly distributed.

If matter was uniformly distributed at the beginning and at the end then you are contradicting thermodynamics.

But like any good scientist you must account for your assumption AND for the alternative hypothesis: which being "matter was not uniformly distributed during the Big Bang".

The 2nd law of thermodynamics says entropy increases with time so if the heat death of the universe is maximum entropy (e.g uniform distribution of matter through the volume of the universe) then at minimum entropy (e.g Big Bang) matter needs not be uniformly distributed.

Yes, there was small quantum fluctuations otherwise nothing could move.